Eyring equation

The general form of the Eyring–Polanyi equation somewhat resembles the Arrhenius equation:

$${\displaystyle \ k={\frac {\kappa k_{\mathrm {B} }T}{h}}e^{-{\frac {\Delta G^{\ddagger }}{RT}}}}$$

where ${\displaystyle k}$ is the rate constant, ${\displaystyle \Delta G^{\ddagger }}$ is the Gibbs energy of activation, ${\displaystyle \kappa }$ is the transmission coefficient, ${\displaystyle k_{\mathrm {B} }}$ is the Boltzmann constant, ${\displaystyle T}$ is the temperature, and ${\displaystyle h}$ is the Planck constant.

The transmission coefficient ${\displaystyle \kappa }$ is often assumed to be equal to one as it reflects what fraction of the flux through the transition state proceeds to the product without recrossing the transition state. So, a transmission coefficient equal to one means that the fundamental no-recrossing assumption of transition state theory holds perfectly. However, ${\displaystyle \kappa }$ is typically not one because (i) the reaction coordinate chosen for the process at hand is usually not perfect and (ii) many barrier-crossing processes are somewhat or even strongly diffusive in nature. For example, the transmission coefficient of methane hopping in a gas hydrate from one site to an adjacent empty site is between 0.25 and 0.5.[1] Typically, reactive flux correlation function (RFCF) simulations are performed in order to explicitly calculate ${\displaystyle \kappa }$ from the resulting plateau in the RFCF. This approach is also referred to as the Bennett-Chandler approach, which yields a dynamical correction to the standard transition state theory-based rate constant.

It can be rewritten as:[2]

$${\displaystyle k={\frac {\kappa k_{\mathrm {B} }T}{h}}e^{\frac {\Delta S^{\ddagger }}{R}}e^{-{\frac {\Delta H^{\ddagger }}{RT}}}}$$

One can put this equation in the following form:

$${\displaystyle \ln {\frac {k}{T}}={\frac {-\Delta H^{\ddagger }}{R}}\cdot {\frac {1}{T}}+\ln {\frac {\kappa k_{\mathrm {B} }}{h}}+{\frac {\Delta S^{\ddagger }}{R}}}$$

where:

• ${\displaystyle k}$ = reaction rate constant
• ${\displaystyle T}$ = absolute temperature
• ${\displaystyle \Delta H^{\ddagger }}$ = enthalpy of activation
• ${\displaystyle R}$ = gas constant
• ${\displaystyle \kappa }$ = transmission coefficient
• ${\displaystyle k_{\mathrm {B} }}$ = Boltzmann constant = R/N_A, N_A = Avogadro constant
• ${\displaystyle h}$ = Planck's constant
• ${\displaystyle \Delta S^{\ddagger }}$ = entropy of activation

If one assumes constant enthalpy of activation, constant entropy of activation, and constant transmission coefficient, this equation can be used as follows: A certain chemical reaction is performed at different temperatures and the reaction rate is determined. The plot of ${\displaystyle \ln(k/T)}$ versus ${\displaystyle 1/T}$ gives a straight line with slope ${\displaystyle -\Delta H^{\ddagger }/R}$ from which the enthalpy of activation can be derived and with intercept ${\displaystyle \ln(\kappa k_{\mathrm {B} }/h)+\Delta S^{\ddagger }/R}$ from which the entropy of activation is derived.

Transition state theory requires a value of the transmission coefficient, called ${\displaystyle \kappa }$ in that theory. This value is often taken to be unity (i.e., the species passing through the transition state ${\displaystyle AB^{\ddagger }}$ always proceed directly to products AB and never revert to reactants A and B). To avoid specifying a value of ${\displaystyle \kappa }$, the rate constant can be compared to the value of the rate constant at some fixed reference temperature (i.e., ${\displaystyle \ k(T)/k(T_{\rm {Ref}})}$) which eliminates the ${\displaystyle \kappa }$ factor in the resulting expression if one assumes that the transmission coefficient is independent of temperature.

Error propagation formulas for ${\displaystyle \Delta H^{\ddagger }}$ and ${\displaystyle \Delta S^{\ddagger }}$ have been published. [3]

• Evans, M.G.; Polanyi M. (1935). "Some applications of the transition state method to the calculation of reaction velocities, especially in solution". Trans. Faraday Soc. 31: 875–894. doi:10.1039/tf9353100875.

• Eyring, H. (1935). "The Activated Complex in Chemical Reactions". J. Chem. Phys. 3 (2): 107–115. Bibcode:1935JChPh...3..107E. doi:10.1063/1.1749604.

• Eyring, H.; Polanyi, M. (2013-11-01). "On Simple Gas Reactions". Zeitschrift für Physikalische Chemie. 227 (11): 1221–1246. doi:10.1524/zpch.2013.9023. ISSN 2196-7156. S2CID 119992451.
• Laidler, K.J.; King M.C. (1983). "The development of Transition-State Theory". J. Phys. Chem. 87 (15): 2657–2664. doi:10.1021/j100238a002.

• Polanyi, J.C. (1987). "Some concepts in reaction dynamics". Science. 236 (4802): 680–690. Bibcode:1987Sci...236..680P. doi:10.1126/science.236.4802.680. PMID 17748308. S2CID 19914017.

• Chapman, S. and Cowling, T.G. (1991). "The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases" (3rd Edition). Cambridge University Press, ISBN 9780521408448

• Eyring equation at the University of Regensburg (archived from the original)
• Online-tool to calculate the reaction rate from an energy barrier (in kJ/mol) using the Eyring equation